Interferometer system

ABSTRACT

An interferometric measurement system capable of measuring tilt of a reflecting surface with respect to a vertical axis. The system preferably includes four laser beams spaced at predetermined distances to measure distances between the measurement system and four locations on the reflecting surface. A controller is also provided for receiving inputs from the measuring laser beams to mathematically determine a tilt of the location being measured.

TECHNICAL FIELD

[0001] The invention relates generally to an interferometer system forposition measurement and more specifically to an interferometer systemand method for improving the accuracy of interferometric measurements.

BACKGROUND ART

[0002] A laser interferometer is often used to accurately measurerelative displacement between two members in a projection exposuresystem used to manufacture semiconductor devices. The laserinterferometer is used as a measuring apparatus for measuring thecoordinates of a wafer stage or mask stage for highly accuratepositioning of a semiconductor wafer or reticle relative to stationaryprojection optics.

[0003] A prior art laser interferometer system is shown in FIGS. 1 and2. The interferometer system typically measures a change in position inmeasurement mirrors 2X and 2Y attached to a movable stage S relative tostationary reference mirrors 1X and 1Y. One or more laser sources (notshown) generate(s) a beam B of light and direct it toward respectivebeam splitters BX and BY. The beam splitters BX and BY split the beams Binto two beams 3 and 4. Beam 3 is the portion of each beam B that isreflected by the beam splitter and directed toward respective referencemirrors 1X and 1Y. The beams 3 reflect off the reference mirrors 1X and1Y and pass through the beam splitters to give beams C. Beam 4 is theportion of each beam B that passes through the beam splitters and isdirected toward respective measurement mirrors 2X and 2Y, and is thenreflected by the measurement mirrors back to the respective beamsplitters. The reflected beams 4 are reflected by the respective beamsplitters where they are combined with reflected beams 3 into thecombined beams C.

[0004] The combined beams C are then directed into respective sensors SXand SY, where they are analyzed to compare the distances represented bybeams 3 and 4. If the measurement mirror 2X moves relative to thereference mirror 1X, the intensity of the combined beam periodicallyincreases and decreases as the reflected light from the two pathsalternately interferes constructively and destructively. Thisconstructive and destructive interference is caused by the two beamsmoving in and out of phase. Each half wavelength of movement of themeasurement mirror results in a total optical path change of onewavelength and thus, one complete cycle of intensity change. The numberof cycle changes indicates the number of wavelengths that themeasurement mirror has moved. Therefore, by counting the number of timesthe intensity of the light cycles between darkest and lightest, thechange in position of the measurement mirror can be estimated as anintegral number of wavelengths.

[0005] Theoretically, if the measurement mirrors 2X and 2Y are perfectlyplanar, and the stage to which they are mounted moves perfectly alongthe x axis, the 2Y mirror surface should not change its position alongthe y axis during x axis movements of the stage, and the beams 3 and 4should stay perfectly in phase as received by the sensor SY. In reality,among other disturbances that may cause interference between beams 3 and4 at the sensor SY in this situation, the mirror 2Y is never perfectlyplanar (of course the same holds true for mirror 2X). In practice, thesemirrors generally have a polishing error of λ/10 or more which equates,for present semiconductor uses, to up to 60 nm deformations measuredfrom the theoretical plane of the mirror surface.

[0006] An example of such a deformation is shown in the 2X mirror inFIG. 2, where the solid line shows the actual deformation of mirror 2X,and the phantom line 2XI shows the ideal perfectly flat surface, withthe deformation indicated by d. The shift of the reflection point fromthe ideal plane, caused by the bowing of the mirror by distance d,brings about a measurement error, since the interferometer is no longermeasuring from the actual reflection point on the ideal planar surface.This error or deformation can be corrected by a pre-measurement of thereflection surface of the measurement mirror 2X. A shift of thereflection point caused by deformation of the mirror in the x-z plane istypically averaged because the stage stroke along the z axis is smallenough, compared with the beam size, to be less significant than theerrors induced by bowing.

[0007] U.S. Pat. No. 5,790,253 to Kamiya describes an interferometersystem for correcting linearity errors of a moving mirror and stage.Thus, Kamiya can correct for the deformation in the mirror along thelong axis of the mirror, which is referred to in the art as correcting“mirror bow”. To correct for mirror bow, Kamiya measures the curvaturedata of the moving mirror prior to its installation on the stage andstores the data as mapping data. Kamiya takes discrete curving errormeasurement along the length of the mirror after it has been mounted onthe stage. Finally, a main controller creates continuous curvature errordata after installation of the mirror on the wafer based on therelationship between the data generated before and after mounting themirror on the stage. The continuous curvature error data is then used ascorrection data for more accurately placing the stage.

[0008] U.S. Pat. No. 5,363,196 to Cameron also describes aninterferometer system for correcting mirror bow of a moving mirrormounted to a stage. Cameron provides two interferometer laser meteringdevices, either one of which is capable of providing measurement data ofthe angle of rotation of the stage in the x-y plane, for use by computercontrolled servo devices that control the x-y movement of the stage. Ina calibration mode, the servo devices may receive data of specificmeasurements defining the respective values of undesired departures fromflatness or straightness of the moving mirror surfaces that are mountedto the stage. The departure data is stored in memory, and may be used bythe computer controlled servo devices to compensate for the undesireddepartures in linearity of the mirrors, during the actual movement andprocessing phases of the stage. Cameron also discloses that, if desired,an additional interferometer may be provided along each of the x and yaxes to measure twist in the moving mirrors. However, because of thelong, narrow aspect ration of both of the moving mirrors, Cameronindicates that determination of twist may not be worth pursuing.

[0009] Sueyoshi, in Japanese HEI9-210648, discloses a method and devicefor measuring a plane shape at a desired pitch by detecting positionalinformation on the plane along three specified points that are separatedby predetermined distances. For example, three x directioninterferometers are aligned in the y direction and spaced atpredetermined distances along the y direction. A similar arrangement isprovide for y direction interferometers. These arrangements are thenused to take measurements for a determination of mirror bowing in the xand y reflective mirrors, respectively.

[0010] As manufacturers of integrated circuits attempt to increasecircuit density and reduce circuit feature size, interferometers arerequired to provide more precise measurement data. As the circuitdensity increases, the tolerance for error in alignment of the stagesystem decreases, so that a shift of the reflection point caused bydeformation of a mirror in the x-z or y-z plane also becomes moresignificant. Additionally, if the stage tilts, a lateral shift of thereflection point occurs which will not be detected by a system forcorrection of mirror bowing. The result is an error in the positionmeasurement of the stage that results in misalignment of circuitpatterns on the wafer (mounted on the stage) relative to one another.

[0011] There is, therefore, a need for an interferometer system thatmeasures and corrects for deformation of moving mirrors as well as tiltof the mirrors and tilt of the stage with respect to the z axis.

SUMMARY OF THE INVENTION

[0012] The invention provides a measuring system that measures andcorrects for deformation and tilt of substantially planar surfaces withrespect to a vertical axis. The measurement system generally includesfirst, second, third and fourth sensors, each capable of generating dataindicative of a distance between the sensors, respectively, andcorresponding locations on a reflective surface of the reflectiveobject. A controller is provided for receiving inputs from the first,second, third and fourth sensors and determining a tilt of thereflective surface with respect to a z axis. A support on which thereflective object is mounted has a generally planar surface that isgenerally perpendicular to the z axis but which may tilt with respectthereto. The reflective object is mounted to the support so that saidreflective surface is in a plane substantially parallel with the z axisand longitudinally extends substantially parallel to an axis normal tothe z axis.

[0013] The first and second sensors are aligned substantially parallelto the axis normal to the z axis along which the reflective surfaceextends longitudinally and are separated by a distance a. The third andfourth sensors are aligned substantially parallel to the axis normal tothe z axis along which the reflective surface extends longitudinally andare separated by the distance a. The first and third sensors are alignedsubstantially parallel to the z axis and are separated by the distancea, and the second and fourth sensors are aligned substantially parallelto the z axis and are separated by the distance a.

[0014] The controller determines a tilt of the reflective surface at alocation ka along the longitudinally extending direction of thereflective surface according to the following formula:

Δ(ka)=Φ((k+1)a)−Φ(ka)

[0015] where:

[0016] Δ(ka) is a measure of a displacement of the reflective surfaceout of the plane substantially parallel with the z axis, at location ka;

[0017] Φ(ka) is a measure of tilt of the reflective surface measured bythe second and fourth sensors; and

[0018] Φ((k+1)a) is a measure of tilt of the reflective surface measuredby the first and third sensors.

[0019] In a preferred embodiment, the first, second, third and fourthsensors comprise first, second, third and fourth measuring laser beamsL1, L2, L3 and L4 that are emitted from an interferometric measurementsystem. The beams L1 and L2 are aligned along an imaginary line parallelto the y axis, and the beams L3 and L4 are aligned along anotherimaginary line parallel to the y axis and separated from the imaginaryline joining L1 and L2 by a distance a. Further, the beams L2 and L4 arealigned along an imaginary line parallel to the z axis, and the beams L1and L3 are aligned along another imaginary line parallel to the z axisand separated from the imaginary line joining L2 and L4 by the distancea. Thus, the distances between L1 and L2, L1 and L3, L2 and L4, and L3and L4 are all equal to a. A reflective surface or mirror can be locatedat position ka along the y axis with respect to the system, and canalternately be located at position ka+a along the y axis with respect tothe system.

[0020] The measurement values obtained by the system through the beamsL1, L2, L3 and L4 when the mirror is at position y=ka are determinedaccording to the following equations:

L 2(ka)=s(ka)+δ(ka)−(a/2)θ(ka)  (4)

L 4(ka)=t(ka)+δ(ka)+(a/2)θ(ka)  (5)

L 1(ka)=s(ka+a)+δ(ka)−(a/2)θ(ka)  (6)

L 3(ka)=t(ka+a)+δ(ka)+(a/2)θ(ka)  (7)

[0021] where:

[0022] θ(x) is a measure of the tilt of the support or stage on whichthe mirror is mounted;

[0023] s(x) is the x coordinate of the shape of the mirror surface atz=0;

[0024] t(x) is the x coordinate of the shape of the mirror surface atz=−a;

[0025] δ(x) is a measurement of the displacement of the stage along thex axis direction that can be due to factors such as vibration, controlerror and the like; and

[0026] a is a distance between measurement beams as defined above.

[0027] In order to simplify terms, J1 is defined as the differencebetween equations (5) and (4), and J2 is defined as the differencebetween equations (7) and (6) as follows:

J 1(ka)≡(L 4(ka)−L 2(ka))/a=(t(ka)−s(ka))/a+θ(ka)=Φ(ka)+θ(ka)  (8)

[0028] where (t(x)−s(x))/a is defined as the mirror tilt Φ(x); and

J 2(ka)≡(L 3(ka)−L1(ka))/a=(t((k+1)a)−s((k+1)a))/a+θ(ka)=Φ((k+1)a)+θ(ka)  (9)

[0029] Next, the difference between terms J2 and J1 is calculated bysubtracting equation (8) from equation (9) to give the mirror tilt atposition y=ka:

Δ(ka)=J 2(ka)−J 1(ka)=Φ((k+1)a)−Φ(ka)  (10)

[0030] After determination of initial tilt values near an end of themirror, the mirror is incrementally moved with respect to the sensors,preferably by at least one motor, to continue measuring displacement ofthe mirror out of the intended plane with respect to the z axis. Themeasurements are performed at each incremental location. In thepreferred embodiment, the mirror is thus moved incrementally in the yaxis direction and measurement values are obtained by the system throughthe beams L1, L2, L3 and L4 at position y=ka+a according to theequations (4)-(7) above, where y now equals ka+a. This process isrepeatedly performed while incrementally moving the mirror in the y axisdirection by a distance a for each iteration. By storing these values ina controller, the mirror tilt can readily be determined at any of theincremental positions along the mirror by a simple summation of thedifferential tilt values obtained from an initial end of the mirror,where the first tilt measurement was made, sequentially up to the actuallocation on the mirror where it is desired to determine the mirror tilt.

[0031] In general, the measurement values determined by theinterferometer system at y=(k−1)a, (k−2)a, . . . a, 0 can be describedas:

Δ((k−1)a)=Φ(ka)−Φ((k−1)a)

Δ((k−2)a)=Φ((k−1)a)−Φ((k−2)a)

. . .

Δ(0)=Φ(a)−Φ(0)

[0032] Since the tilt measurement values are differential values definedwith respect to the previously measured value, a tilt value for a givenlocation can be determined by summing the sequence of values precedingand including that location. A summation of the general equations givenabove gives: $\begin{matrix}{{\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}} = {{\Phi ({ka})} - {\Phi (0)}}} & (11)\end{matrix}$

[0033] Therefore, a mirror tilt value at position ka is given by theequation: $\begin{matrix}{{\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}}}} & (12)\end{matrix}$

[0034] Thus, the mirror tilt of the mirror at position ka can be readilydetermined using equation (12). This value is then added, by thecontroller, to the interferometer position measurement value that isinputted to the controller, to more accurately position the stage byincluding the lateral offset due to the mirror tilt.

[0035] Additionally, the measurement system preferably includes a secondreflective surface and a second set of four sensors, preferably fourmeasuring laser beams incorporated into an interferometric measurementsystem that measures the second substantially planar reflective surfacethat is oriented orthogonally to the first reflective surface.Measurements from the second set of sensors are also inputted to thecontroller for a determination of tilt of the second reflective surface.The determination is made according to the same procedure used todetermine tilt of the first reflective surface.

[0036] The invention also provides a method of measuring the tilt of asubstantially planar surface that includes (1) providing a measurementsystem having the capability of measuring distances between first,second, third and fourth adjacent locations on the substantially planarsurface and respective first, second, third and fourth adjacentlocations on the measurement system, where the distances measured arealong imaginary lines substantially perpendicular to the substantiallyplanar surface; (2) positioning the substantially planar surfaces suchthat the measurement system is near an end of the substantially planarsurface; (3) measuring distances between the pairs of first, second,third and fourth locations; (4) subtracting the distance between thesecond locations from the distance between the fourth locations anddividing the difference by a distance between the second and fourthlocations on the substantially planar surface to give a term J1; (5)subtracting the distance between the first locations from the distancebetween the third locations and dividing the difference by the distancebetween the first and third locations on the substantially planarsurface to give a term J2; and (6) determining a tilt of thesubstantially planar surface at the location of the substantially planarsurface according to the following formula:

Δ(ka)=J 2(ka)−J 1(ka)=Φ((k+1)a)−Φ(ka)

[0037] where:

[0038] Δ(ka) is a measure of a displacement out of the substantiallyplanar surface;

[0039] Φ(ka) is a measure of tilt of the substantially planar surfacewith respect to the vertical axis measured between the second and fourthlocations;

[0040] Φ((k+1)a) is a measure of tilt of the substantially planarsurface with respect to the vertical axis measured between the first andthird locations; and

[0041] a is a distance between the first and second locations on thesubstantially planar surface.

[0042] The method preferably further includes (1) incrementally movingthe substantially planar surface in a direction parallel to an axisnormal to the vertical axis and away from the end of the surface by thedistance a; (2) measuring distances between the first, second, third andfourth locations on the measurement system and the respective four newlocations on the substantially planar surface; (3) subtracting thedistance between the second locations from the distance between thefourth locations and dividing the difference by a distance between thesecond and fourth locations on the substantially planar surface to givea term J1; (4) subtracting the distance between the first locations fromthe distance between the third locations and dividing the difference bythe distance between the first and third locations on the substantiallyplanar surface to give a term J2; and (5) determining a tilt of thesubstantially planar surface at the new location incrementally removedfrom a previously measured location according to the following formula:

Δ(ka+a)=J 2(ka+a)−J 1(ka+a)=Φ((k+2)a)−Φ((k+1)a)

[0043] where:

[0044] Δ(ka+a) is a measure of a displacement out of the substantiallyplanar surface;

[0045] Φ((k+1)a) is a measure of tilt of the substantially planarsurface with respect to the vertical axis measured between the secondand fourth locations on the measurement system and the new locations onthe substantially planar surface; and

[0046] Φ((k+2)a) is a measure of tilt of the substantially planarsurface with respect to the vertical axis measured between the first andthird locations on the measurement system and the new locations on thesubstantially planar surface.

[0047] Incremental measurements are continued by incrementally repeatingthe previously described incremental procedure, until an opposite end ofthe substantially planar surface is reached and no further incrementalmeasurements can be taken, or until a predetermined length of thesubstantially planar surface has been measured.

[0048] As described above, a determination of the tilt of thesubstantially planar surface with respect to the vertical axis for anypredetermined position ka can be determined according to the followingformula:${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}}}$

[0049] where:

[0050] Φ(ka) is a measure of tilt of the substantially planar surfacewith respect to the vertical axis at position ka;

[0051] Φ(0) is a measure of tilt of the substantially planar surfacenear the end of the substantially planar surface where an initialmeasurement was taken; and

[0052] Δ(ma) is a measure of displacement out of the substantiallyplanar surface, at locations where m=0, 1, 2, . . . k−1.

[0053] The above is a brief description of some deficiencies in theprior art and advantages of the invention. Other features, advantages,and embodiments of the invention will be apparent to those skilled inthe art from the following description, drawings and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

[0054]FIG. 1 is a plan view of a prior art interferometer system formeasuring position of a stage movable in x and y directions;

[0055]FIG. 2 is a partial view of the prior art interferometer system ofFIG. 1 showing an exaggerated view of mirror bowing, or deformation inthe x-y plane;

[0056]FIG. 3 is a side view of an interferometer system where a movingmirror is perfectly parallel to the z axis.

[0057]FIG. 4 is a side view of an interferometer system where a movingmirror tilts at an angle Φ with respect to the z axis.

[0058]FIG. 5 is a schematic showing changes in optical path lengths oflight beams due to mirror tilt when no stage tilt is present;

[0059]FIG. 6 is a schematic showing changes in optical path lengths oflight beams due to combined effect of mirror tilt and stage tilt;

[0060]FIG. 7 is a perspective view of an interferometer system of theinvention applied to a wafer stage of a projection type exposureapparatus;

[0061]FIG. 8 is a schematic showing the pattern of measurement beamsresultant from a preferred layout of interferometers used to measuremirror deformation in the x-z plane; and

[0062]FIG. 9 is a schematic showing a structure that enables a stage ofan interferometer system to be incrementally moved.

[0063] Corresponding reference characters indicate corresponding partsthroughout the several views of the drawings.

DESCRIPTION OF THE INVENTION

[0064] An interferometer, such as one used in the prior art system 10shown in FIGS. 1-2, is used to accurately measure the displacement of ameasurement target (e.g., stage S) by using interference between lightwaves that have propagated along a measurement optical path 4 and areference optical path 3. The interferometer may be used as a positionmeasurement system of a stage apparatus such as a wafer stage or a maskstage in a one-shot or scan type exposure apparatus for which highlyprecise driving control is required. The interferometer is not limitedto use with an exposure apparatus. The interferometer may be used toaccurately measure the relative displacement between two members invarious high precision tools, for example.

[0065] A measurement mirror (reflector) 2X is attached to the stage Sand movable therewith to provide measurement of the measurement opticalpath 4, and a reference mirror (reflector) 1X is attached to a lens orother stationary portion of the exposure apparatus to provide thereference optical path 3. The measurement mirror 2X is attached to thestage S parallel to the y direction and measurement mirror 2Y isattached to the stage S parallel to the x direction (FIG. 1). Themeasurement mirror 2X is used to measure displacement of the stage alongthe x axis while the measurement mirror 2Y is used to measuredisplacement of the stage along the y axis.

[0066] As shown in exaggerated form in FIG. 2, a measurement mirror, inthis case measurement mirror 2X, is never perfectly planar but will havea certain amount of “bow” or deformation with respect to the axis thatit is set up to parallel, in this case, the y axis. The deformation outof the ideal plane 2XI that is parallel to the y axis, results in achange in the length of the measuring beam 4 as the stage S is moved inthe y direction and the area of the mirror having the deformation dpasses the path of the beam 4. Unless this deviation is compensated for,the deformation d will cause an error of the stage position. Theseinaccuracies in the stage position cause misalignment of the circuits onthe semiconductor wafer that is mounted on the stage S.

[0067] The interferometer system shown in FIGS. 1 and 2 cannot detectdeviations of the measurement mirror 2X where the surface of themeasurement mirror 2X deviates from the ideal surface that is parallelto the z axis. It is known that a measurement mirror, in this casemeasurement mirror 2X, is never perfectly parallel to the z axis, butwill have a certain amount of “tilt” with respect to the z axis. FIG. 3shows the case where the moving mirror is perfectly parallel to the zaxis. FIG. 4 shows the case where the moving mirror tilts at an angle Φwith respect to the z axis. When the moving mirror tilts at an angle ofΦ with respect to the z axis, and the interferometer has an offsetdistance of h, the interferometer measurement error is Δx, where Δx=hΦ.For example, when h=1 mm and Φ=5 μrad, Δx=5 nm. The tilt angle Φ is afunction of the stage position x or y as well as the mirror bow d. If Φcan be measured and mapped as a function of x or y, the measurementerror Δx can be corrected using the equation Δx=hΦ.

[0068] The interferometer systems of the invention measure displacementdue to mirror tilt and stage tilt, as well as displacement due to mirrorbowing. In order to measure tilt, measurement beams must be displacedwith regard to one another along the z axis, as shown in FIG. 5 by theplacement of measurement beams L1 and L3. FIG. 5 is a partial schematicview of an interferometer system 20 according to the invention. As canbe seen in FIG. 5, beams L1 and L3 are displaced with regard to oneanother along the z axis, while being aligned with regard to the y axis.Thus, any differences in the measurement lengths of beams L1 and L3 willindicate tilt with respect to the z axis and will not reflect bowingdeviations. As is evident in FIG. 5, the tilt of the surface ofmeasurement mirror 26 x with regard to the ideal 26XI by the angle Φresults in a difference between the lengths of measurement beams L1 andL3 that when analyzed can give the degree of tilt between the twomeasuring points contacted by measurement beams L1 and L3, respectively.

[0069] When the stage 36 is not tilted with respect to the z axis (FIG.5), the coordinates of a point on the reflecting surface of mirror 26Xupon which a measurement beam is incident are given by the equation (1):

z=−(1/tan Φ)(x+L)  (1)

[0070] where:

[0071] z=the z coordinate of the location on the measurement mirror 26Xon which the measurement beam is incident;

[0072] Φ=the mirror tilt with respect to the z axis, as described above;

[0073] x=the x coordinate of the location on the measurement mirror 26Xon which the measurement beam is incident; and

[0074] L=the distance between the center of tilt (i.e., the central axisof the illumination lens, or the optical axis of the projection lens)and the location on the measurement mirror 26X on which the measurementbeam is incident, measured parallel to the x axis.

[0075] In situations where the stage 36 is tilted with respect to the zaxis (FIG. 6), the mirror surface becomes laterally shifted in the xaxis direction by an amount in addition to the mirror tilt, due to thetilting angle θ of the stage 36 about the z axis. The additional lateraldisplacement is indicated by Δx in FIG. 6 and is measured by thedisplacement between the x coordinate of the mirror surface when stagetilt is zero (as indicated by a phantom line in FIG. 6) and the xcoordinate of the mirror surface as effected by both the mirror tilt andthe stage tilt (as shown by the solid line). The coordinates of a pointon the reflecting surface of mirror 26X upon which a measurement beam isincident, in a situation as shown in FIG. 6, are given by the equation(2):

x(tan Φ sin θ−cos θ)−z(tan Φ cos θ+sin θ)+L=0  (2)

[0076] where θ=the tilting angle θ of the stage 36 about the z axis, asdescribed above.

[0077] By setting the axis of L1 of the interferometer to have thecoordinate z=0, the deviation Δx of the mirror surface where itintersects with L1 can be derived. By setting z=0 in equations (1) and(2) above and taking the difference in the x values arrived at, anequation for Δx is defined as follows:

Δx=Lθ ²/2+LΦθ  (3)

[0078] Thus, the second term in the equation (3) takes into account themirror tilt Φ and the first term is only affected by stage tilt θ. As anexample, if the mirror tilt Φ is about 3 μrad, stage tilt θ is about 1mrad and L is about 500 mm, then the component effected by mirror tilt Φ(i.e., LΦθ) alters the term Δx by about 1.5 nm. Accordingly, if themirror tilt Φ is not measured and corrected, in this example theinterferometer measurement will have a 1.5 nm measurement error.Although in the past this error has been ignored because it is muchsmaller than the error values due to mirror bowing, the density ofintegrated circuit patterns have become such that the tilt error can nolonger be ignored.

[0079]FIG. 7 shows a preferred arrangement of an interferometer system20 according to the invention applied to a wafer stage 36 of aprojection type exposure apparatus 100. The apparatus is described infurther detail below. FIG. 8 shows the relative positioning ofmeasurement beams L1, L2, L3 and L4 emitted from interferometer system30X that has the ability to measure mirror tilt of the measurementmirror 26X with respect to the z axis.

[0080]FIG. 8 shows the relational positions of beams L1, L2, L3 and L4in solid lines when the mirror 26X is located at position ka along the yaxis with respect to the interferometer system 30X, and the relationalpositions of beams L1, L2, L3 and L4 in phantom lines when the mirror26X is located at position ka+a along the y axis with respect to theinterferometer system. The beams L1 and L2 are aligned along animaginary line parallel to the y axis, and the beams L3 and L4 arealigned along another imaginary line parallel to the y axis andseparated from the imaginary line joining L1 and L2 by a distance a.Further, the beams L2 and L4 are aligned along an imaginary lineparallel to the z axis, and the beams L1 and L3 are aligned alonganother imaginary line parallel to the z axis and separated from theimaginary line joining L2 and L4 by a distance a. Thus, the distancesbetween L1 and L2, L1 and L3, L2 and L4, and L3 and L4 are all equal toa.

[0081] The measurement values obtained by the interferometer systemthrough the beams L1, L2, L3 and L4 at position y=ka are determinedaccording to the following equations:

L 2(ka)=s(ka)+δ(ka)−(a/2)θ(ka)  (4)

L 4(ka)=t(ka)+δ(ka)+(a/2)θ(ka)  (5)

L 1(ka)=s(ka+a)+δ(ka)−(a/2)θ(ka)  (6)

L 3(ka)=t(ka+a)+δ(ka)+(a/2)θ(ka)  (7)

[0082] where:

[0083] θ(x) is a measure of the stage 36 pitching as described above;

[0084] s(x) is the x coordinate of the shape of the mirror 26X surfaceat z=0;

[0085] t(x) is the x coordinate of the shape of the mirror 26X surfaceat z=−a;

[0086] δ(x) is a measurement of the displacement of the stage 36 alongthe x axis direction that can be due to factors such as vibration,control error and the like; and

[0087] a is a distance between measurement beams as defined above.

[0088] In order to simplify terms, J1 is defined as the differencebetween equations (5) and (4), and J2 is defined as the differencebetween equations (7) and (6) as follows:

J 1(ka)≡(L 4(ka)−L 2(ka))/a=(t(ka)−s(ka))/a+θ(ka)=Φ(ka)+θ(ka)  (8)

[0089] where (t(x)−s(x))/a is defined as the mirror tilt Φ(x); and

J 2(ka)≡(L 3(ka)−L1(ka))/a=(t((k+1)a)−s((k+1)a))/a+θ(ka)=Φ((k+1)a)+θ(ka)  (9)

[0090] Next, the difference between terms J2 and J1 is calculated bysubtracting equation (8) from equation (9) to give the mirror tilt atposition y=ka:

Δ(ka)=J 2(ka)−J 1(ka)=Φ((k+1)a)−Φ(ka)  (10)

[0091] The mirror 26X is next moved incrementally in the y axisdirection. FIG. 9 illustrates the structure that enables the stage S(and therefore the mirror 26X) to be moved. Accordingly, measurementvalues are obtained by the interferometer system through the beams L1,L2, L3 and L4 at position y=ka+a (as shown by the phantom lines in FIG.8) according to the equations (4)-(7) above, where y now equals ka+a:

L 2(ka+a)=s(ka+a)+δ(ka+a)−(a/2)θ(ka+a)  (4′)

L 4(ka+a)=t(ka+a)+δ(ka+a)+(a/2)θ(ka+a)  (5′)

L 1(ka+a)=s(ka+a+a)+δ(ka+a)−(a/2)θ(ka+a)  (6′)

L 3(ka+a)=t(ka+a+a)+δ(ka+a)+(a/2)θ(ka+a)  (7′)

[0092] Again, in order to simplify terms, J1 and J2 are definedaccording to the equations (8) and (9) above:

J 1((k+1)a)≡(L 4((k+1)a)−L 2((k+1)a))/a=(t((k+1)a)−s((k+1)a))/a+θ((k+1)a)=Φ((k+1)a)+θ((k+1)a)  (8′)

J 2((k+1)a)≡(L 3((k+1)a)−L1((k+1)a))/a=(t((k+2)a)−s((k+2)a))/a+θ((k+1)a)=Φ((k+2)a)+θ((k+1)a)  (9′)

[0093] Again, the difference between terms J2 and J1 is calculated bysubtracting equation (8′) from equation (9′) to give the mirror tilt atposition x=ka+a:

Δ((k+1)a)=J 2((k+1)a)−J 1((k+1)a)=Φ((k+2)a)−Φ((k+1)a)  (10′)

[0094] This process is repeatedly performed while incrementally movingthe mirror 26X in the y axis direction by a distance a for eachiteration. By storing these values in the controller 40, the mirror tiltcan readily be determined at any of the incremental positions along themirror 26X by a simple summation of the differential tilt valuesobtained from an initial end of the mirror 26X, where the first tiltmeasurement was made, sequentially up to the actual location on themirror 26X where it is desired to determine the mirror tilt.

[0095] In general, the measurement values determined by theinterferometer system 30X at y=(k−1)a, (k−2)a, . . . a, 0 can bedescribed as:

Δ((k−1)a)=Φ(ka)−Φ((k−1)a)

Δ((k−2)a)=Φ((k−1)a)−Φ((k−2)a)

. . .

Δ(0)=Φ(a)−Φ(0)

[0096] Since the tilt measurement values are differential values definedwith respect to the previously measured value, a tilt value for a givenlocation can be determined by summing the sequence of values precedingand including that location. A summation of the general equations givenabove gives: $\begin{matrix}{{\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}} = {{\Phi ({ka})} - {\Phi (0)}}} & (11)\end{matrix}$

[0097] Therefore, a mirror tilt value at position ka is given by theequation: $\begin{matrix}{{\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta \quad ({ma})}}}} & (12)\end{matrix}$

[0098] Thus, the mirror tilt of the mirror 26X can be readily determinedusing equation (12). If we assume that the initial measurement locationalong mirror 26X is a location very near the intersection of the mirrors26X and 26Y shown in FIG. 7, then the incremental measurements will bemade moving toward the opposite end of the mirror 26X along the y axis,as schematically illustrated by FIG. 8 which shows a one step increment.It is not possible to directly measure the tilt at the very ends of themirror, due to the lateral displacement of the beams L1 and L3 from L2and L4, and because all of the beams L1-L4 are used to measure a tiltvalue, as described above. If a mirror area L is used for exposure, amirror length of L+2a can be used to measure mirror tilt.

[0099] Further assuming that mirror 26X is located in position ka inFIG. 7, the tilt of the mirror 26X at position ka can be determined bysumming all of the incremental differential values of Δ and adding themto the initial measurement, as described above in equation (12). Thisvalue is then added, by the controller 40, to the interferometerposition measurement value that is inputted to the controller 40, tomore accurately position the stage 36 by including the lateral offsetdue to the mirror tilt. It is noted that although only incrementalvalues for the mirror tilt are measured and stored, it is possible tointerpolate, using algorithms known in the art (such as a Splineinterpolation algorithm), between values to determine a value of a finalincrement for a position of the stage that is not located directly atone of the incremental measurement positions. If a mirror area L is usedfor exposure, a mirror length of L+2a can be used to measure mirrortilt.

[0100] In a similar manner to that described above, determination of thetilt measurements of the mirror 26Y can be determined, and inputted tothe controller 40. The tilt measurement value is added by the controller40 to the interferometer position measurement value of the y position ofthe stage, which is inputted to the controller 40 from interferometersystem 30Y, to more accurately position the stage 36 by including thelateral offset in the y direction due to the mirror tilt (i.e., thecomponent of Δy effected by mirror tilt in the y-z plane).

[0101] It is further noted that interferometer systems 30X and 30Y havethe capability of correcting for mirror bowing, and providing controlmeasurements to control yawing, according to techniques known in theart. Beams L1, L2, L3 and L4 in each of interferometer systems 30X and30Y may be formed using a single laser with appropriate opticalcomponents to form the four beams, as would be readily apparent to oneof ordinary skill in the art, or may comprise four aligned sources toform the four beams, or two beams with appropriate optical components,or other combinations that would be apparent to those of ordinary skillin the art. Preferably a single laser source is used in each ofinterferometer systems 30X and 30Y. Further, although laser metering ispreferred to precisely measure the distances according to the invention,the invention is not to be limited only to the use of laser metering,but may be applied to other distance measuring technologies or sensorsthat are capable of providing an adequate degree of measurementprecision for the use to which the invention is being applied.

[0102] Referring again to FIG. 7, a schematic illustration of an exampleof an interferometric measuring system 20 according to the invention isshown applied to an exposure apparatus, the entire apparatus beinggenerally referred to by reference numeral 100. The exposure apparatus100 generally comprises an illumination system 102, the wafer stage 36for supporting and positioning the wafer 34, a reticle stage (not shown)for supporting and positioning a reticle 104, and motors (not shown) forpositioning the wafer stage 36 and the reticle stage. The illuminationsystem 102 projects energy in the form of, for example, light or anelectron beam light through a mask pattern (e.g., circuit pattern for asemiconductor device) formed in the reticle 104 that is supported andscanned using the reticle stage. In an optical system, the illuminationsystem 102 is part of an optical system that further includes aprojection lens 106. The illumination system 102 preferably has anoptical integrator (not shown) for producing secondary light sourceimages and a condenser lens for illuminating the reticle 104 withuniform light flux. The projection lens 106 focuses the light orelectron beam received through the reticle 104, onto the wafer 34. Thewafer 34 is positioned under the projection lens 106 and preferablyheld, for example, by vacuum suction or electrostatic suction on a waferholder (not shown) that is supported by the wafer stage 36. Inoperation, light or electron beams from the illumination system 102 passthrough the reticle 104 and expose resist on the wafer 34, which issupported and scanned using the wafer stage 36 driven by the motor.

[0103] The stage 36 is movable in at least two directions along the xand y axes in a plane perpendicular to an optical axis of the exposureapparatus 200, which is defined as the z axis. Measurement mirrors 26Xand 26Y are provided at two locations around the stage 36. Themeasurement mirror 26Y has its reflecting surface extending along orparallel to the x axis for measuring movement of the stage 36 in the ydirection and the measurement mirror 26X has its reflecting surfaceextending along or parallel to the y axis for measuring movement of thestage in the x direction. The interferometer systems 30X and 30Y aremounted with respect to the mirrors 26X and 26Y, respectively, so as toilluminate the respective mirrors with a pattern of measuring beams L1,L2, L3 and L4, as described above. In the system shown in FIG. 7, eachof the interferometer systems 30X and 30Y is provided with a laser head38X, 38Y that provides the laser illumination for the formation of themeasurement beams L1, L2, L3 and L4. The interferometer systems 30X and30Y input measurement values to the controller 40, which analyzes theinputted measurement data and determines an accurate position of thestage 36, as described above. In addition to measurement of distancevalues in the X and Y directions, the interferometer systems 30X and30Y, together with the controller 40, can provide accurate yaw, pitchand roll control, as described above, where pitch is defined here astilt in the x-z plane and roll is defined as tilt in the y-z plane. Toprovide more accurate measurement and positioning, the interferometersystems incrementally measure the tilt of the mirrors 26X and 26Y, asdescribed above, and these values are stored in the controller to berelied upon, during real time measurement, to more accurately measurelateral offset Δx and Δy, but accounting for the influence of theseoffset values that is attributed to mirror tilt. Additionally, theinterferometer systems may be used to incrementally measure mirror bow,according to methods known in the art, and these values can also bestored by the controller 40 to be combined with real time measurementsto more accurately determine the position of the stage.

[0104] Further details of the components of the exposure apparatus 100may be referenced from U.S. Pat. No. 5,528,118 to M. Lee, for example.The entire contents of U.S. Pat. No. 5,528,118 are hereby incorporatedby reference thereto. It is to be understood that the invention is notlimited to the exposure apparatus 100 described herein or to exposuresystems for wafer processing. The general reference to the exposureapparatus 100 is purely for illustrating an embodiment in an environmentin which the invention may be used.

[0105] It will be observed from the above description that theinterferometer systems of the invention provide a number of advantagesover prior art systems. Importantly, the interferometer systems accountfor displacement of the measurement mirrors due to a compound effect ofmirror tilt and stage tilt, thus providing more accurate stage positionmeasurement than possible with prior art interferometer systems.

What is claimed is:
 1. A measurement system for determining the tilt ofa reflective object mounted to a support, the system comprising: first,second, third and fourth sensors, each capable of generating dataindicative of a distance between the first, second, third or fourthsensor, respectively, and a reflective surface of the reflective object;and a controller for receiving inputs from the first, second, third andfourth sensors and determining a tilt of the reflective surface withrespect to a z axis;  wherein: the support has a generally planarsurface that is generally perpendicular to the z axis but which may tiltwith respect thereto, the reflective object is mounted to the support sothat the reflective surface is in a plane substantially parallel withthe z axis and longitudinally extends substantially parallel to an axisnormal to the z axis; the first and second sensors are alignedsubstantially parallel to the axis normal to the z axis along which thereflective surface extends longitudinally and are separated by adistance a; the third and fourth sensors are aligned substantiallyparallel to the axis normal to the z axis along which the reflectivesurface extends longitudinally and are separated by the distance a; thefirst and third sensors are aligned substantially parallel to the z axisand are separated by a distance b; the second and fourth sensors arealigned substantially parallel to the z axis and are separated by thedistance b; and the controller determines a tilt of the reflectivesurface at a location ka along the longitudinally extending direction ofthe reflective surface according to the following formula:Δ(ka)=Φ((k+1)a)−Φ(ka)  where: Δ(ka) is a measure of a displacement ofthe reflective surface out of the plane substantially parallel with thez axis, at location ka; Φ(ka) is a measure of tilt of the reflectivesurface measured by the second and fourth sensors; and Φ((k+1)a) is ameasure of tilt of the reflective surface measured by the first andthird sensors.
 2. The measurement system of claim 1, wherein the thirdand fourth sensors are aligned substantially parallel to the axis normalto the z axis along which the reflective surface extends longitudinallyand are separated by a distance c.
 3. The measurement system of claim 1,wherein: θ(x) is a measure of tilt of the support; s(x) is a measure ofdisplacement, out of the plane substantially parallel with the z axis,of the reflective surface when z=0; t(x) is a measure of displacement,out of the plane substantially parallel with the z axis, of thereflective surface when z=b; δ(x) is a measure of displacement of thesupport along the x axis normal to the z axis; a measurement value forthe second sensor when the reflective surface is at a position y=ka isdetermined by L2(ka)=s(ka)+δ(ka)−(b/2)θ(ka); a measurement value for thefourth sensor when the reflective surface is at a position y=ka isdetermined by L4(ka)=t(ka)+δ(ka)+(b/2)θ(ka); a measurement value for thefirst sensor when the reflective surface is at a position y=ka isdetermined by L1(ka)−s(ka+a)+δ(ka)−(b/2)θ(ka); a measurement value forthe third sensor when the reflective surface is at a position y=ka isdetermined by L3(ka)=t(ka+a)+δ(ka)+(b/2)θ(ka);Δ(ka)=J2(ka)−J1(ka)=Φ((k+1)a)−Φ(ka), whereJ1(ka)≡(L4(ka)−L2(ka))/b=(t(ka)−s(ka))/bθ(ka)=Φ(ka)+θ(ka), andJ2(ka)≡(L3(ka)−L1 (ka))/b=(t((k+1)a)−s((k+1)a))/b+θ(ka)=Φ((k+1)a)+θ(ka);the reflective surface can be moved to a position y=ka+a along the yaxis normal to the z axis along which the reflective surface extendslongitudinally; a measurement value for the second sensor when thereflective surface is at a position y=ka+a is determined byL2(ka+a)=s(ka+a)+δ(ka+a)−(b/2)θ(ka+a); a measurement value for thefourth sensor when the reflective surface is at a position y=ka+a isdetermined by L4(ka+a)=t(ka+a)+δ(ka+a)+(b/2)θ(ka+a); a measurement valuefor the first sensor when the reflective surface is at a position y=ka+ais determined by L1(ka+a)=s(ka+a+a)+δ(ka+a)−(b/2)θ(ka+a); a measurementvalue for the third sensor when the reflective surface is at a positiony=ka+a is determined by L3(ka+a)=t(ka+a+a)+δ(ka+a)+(b/2)θ(ka+a);Δ((k+1)a)=J2((k+1)a)−J1((k+1)a)=Φ((k+2)a)−Φ((k+1)a), whereJ1((k+1)a)≡(L4((k+1)a)−L2((k+1)a))/b=(t((k+1)a)−s((k+1)a))/b+θ((k+1)a)=Φ((k+1)a)+θ((k+1)a),andJ2((k+1)a)≡(L3((k+1)a)=L1((k+1)a))/b=(t((k+2)a)−s((k+2)a))/b+θ((k+1)a)=Φ((k+2)a)+θ((k+1)a);the reflective surface can be incrementally moved to additionalpositions in multiples of a along the axis normal to the z axis alongwhich the reflective surface extends longitudinally and additionalmeasurement values for the sensors can be determined to arrive at a setof measurement values {Δ((k−1)a)=Φ(ka)−Φ((k−1)a);Δ((k−2)a)=Φ((k−1)a)−Φ((k−2)a); . . . ; Δ(0)=Φ(a)−Φ(0)}; and thecontroller determines a summation of the set of measurement values as${\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}} = {{\Phi ({ka})} - {\Phi (0)}}$

and a tilt of the reflective surface at a location ka along thelongitudinally extending direction of the reflective surface as${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{{\Delta ({ma})}.}}}$


4. The measurement system of claim 1, wherein the first, second thirdand fourth sensors comprise first, second, third and fourth laser beams.5. The measurement system of claim 4, wherein the first second, thirdand fourth laser beams are incorporated into an interferometer system.6. The measurement system of claim 1, wherein the reflective surfacecomprises a first reflective surface; the measurement system furthercomprising: fifth, sixth, seventh and eighth sensors, each capable ofgenerating data indicative of a distance between the fifth, sixth,seventh or eighth sensor, respectively, and a second surface thatcorresponds to a reflective surface of a second reflective object;wherein the controller receives inputs from the fifth, sixth, seventhand eighth sensors and determines a tilt of the second reflectivesurface with respect to the z axis, wherein the second reflective objectis mounted to the support so that the second reflective surface is in asecond plane substantially parallel with the z axis and longitudinallyextends substantially parallel to an axis normal to the z axis andnormal to the axis which the first reflective surface extendssubstantially parallel to; wherein the fifth and sixth sensors arealigned substantially parallel to the axis normal to the z axis alongwhich the second reflective surface extends longitudinally and areseparated by a distance a; wherein the seventh and eighth sensors arealigned substantially parallel to the axis normal to the z axis alongwhich the second reflective surface extends longitudinally and areseparated by the distance a; wherein the fifth and seventh sensors arealigned substantially parallel to the z axis and are separated by thedistance b; wherein the sixth and eighth sensors are alignedsubstantially parallel to the z axis and are separated by the distanceb; and wherein the controller determines a tilt of the second reflectivesurface at a location ka along the longitudinally extending direction ofthe second reflective surface according to the following formula:Δ(ka)=Φ((k+1)a)−Φ(ka)  where: Δ(ka) is a measure of a displacement ofthe second reflective surface out of the second plane substantiallyparallel with the z axis, at location ka; Φ(ka) is a measure of tilt ofthe second reflective surface measured by the sixth and eighth sensors;and Φ((k+1)a) is a measure of tilt of the reflective surface measured bythe fifth and seventh sensors.
 7. The measurement system of claim 6,wherein: θ(y) is a measure of tilt of the support; s(y) is a measure ofdisplacement, out of the plane substantially parallel with the z axis,of the second reflective surface when z=0; t(y) is a measure ofdisplacement, out of the plane substantially parallel with the z axis,of the second reflective surface when z=−b; δ(y) is a measure ofdisplacement of the support along the y axis normal to the z axis; ameasurement value for the sixth sensor when the second reflectivesurface is at a position x=ka is determined byL6(ka)=s(ka)+δ(ka)−(a/2)θ(ka); a measurement value for the eighth sensorwhen the second reflective surface is at a position x=ka is determinedby L8(ka)=t(ka)+δ(ka)+(a/2)θ(ka); a measurement value for the fifthsensor when the second reflective surface is at a position x=ka isdetermined by L5(ka)=s(ka+a)+δ(ka)−(a/2)θ(ka); a measurement value forthe seventh sensor when the second reflective surface is at a positionx=ka is determined by L7(ka)=t(ka+a)+δ(ka)+(a/2)θ(ka);Δ(ka)=J2(ka)−J1(ka)=Φ((k+1)a)−Φ(ka), whereJ1(ka)≡(L8(ka)−L6(ka))/b=(t(ka)−s(ka))/b+θ(ka)=Φ(ka)+θ(ka), andJ2(ka)≡(L7(ka)−L5(ka))/b=(t((k+1)a)−s((k+1)a))/b+θ(ka)−Φ((k+1)a)+θ(ka);the second reflective surface can be moved to a position x=ka+a alongthe axis normal to the z axis along which the second reflective surfaceextends longitudinally; a measurement value for the sixth sensor whenthe second reflective surface is at a position x=ka+a is determined byL6(ka+a)=s(ka+a)+δ(ka+a)−(a/2)θ(ka+a); a measurement value for theeighth sensor when the second reflective surface is at a position x=ka+ais determined by L8(ka+a)=t(ka+a)+δ(ka+a)+(a/2)θ(ka+a); a measurementvalue for the fifth sensor when the second reflective surface is at aposition x=ka+a is determined byL5(ka+a)=s(ka+a+a)+δ(ka+a)−(a/2)θ(ka+a); a measurement value for theseventh sensor when the second reflective surface is at a positionx=ka+a is determined by L7(ka+a)=t(ka+a+a)+δ(ka+a)+(a/2)θ(ka+a);Δ((k+1)a)=J2((k+1)a)−J1((k+1)a)=Φ((k+2)a)−Φ((k+1)a), whereJ1((k+1)a)≡(L8((k+1)a)−L6((k+1)a))/b=(t((k+1)a)−s((k+1)a))/b+θ((k+1)a)=Φ((k+1)a)+θ((k+1)a),andJ2((k+1)a)≡(L7((k+1)a)−L5((k+1)a))/b=(t((k+2)a)−s((k+2)a))/b+θ((k+θ((k+1)a)−Φ((k+2)a)+θ((k+1)a);the second reflective surface can be incrementally moved to additionalpositions in multiples of a along the axis normal to the z axis alongwhich the second reflective surface extends longitudinally andadditional measurement values for the sensors can be determined toarrive at a set of measurement values {Δ((k−1)a)=Φ(ka)−Φ((k−1)a);Δ((k−2)a)=Φ((k−1)a)−Φ((k−2)a); . . . ; Δ(0)=Φ(a)−Φ(0)}; and thecontroller determines a summation of the set of measurement values as${\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}} = {{\Phi ({ka})} - {\Phi (0)}}$

and a tilt of the second reflective surface at a location ka along thelongitudinally extending direction of the second reflective surface as${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{{\Delta ({ma})}.}}}$


8. The measurement system of claim 6, wherein said first, second thirdand fourth sensors comprise first, second, third and fourth laser beams;and wherein said fifth, sixth, seventh and eighth sensors comprisefifth, sixth, seventh and eighth laser beams.
 9. The measurement systemof claim 8, wherein said first second, third and fourth laser beams areincorporated into a first interferometer system, and wherein said fifth,sixth, seventh and eighth laser beams are incorporated into a secondinterferometer system.
 10. The measurement system of claim 1, furthercomprising at least one motor operatively mounted to said support tomove said support, said reflective surface and said reflective object inthe directions along which said reflective surface longitudinallyextends, wherein said motor incrementally moves said reflective surfaceto measure displacement of said reflective surface out of said planesubstantially parallel with the z axis at each incremental location. 11.The measurement system of claim 10, wherein said tilt of said reflectivesurface, at a position ka is defined by:${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}}}$

where: Φ(ka) is a measure of tilt of said reflective surface at positionka; Φ(0) is a measure of tilt of said reflective surface at an initialmeasurement location near one end of said reflective surface: and Δ(ma)is a measure of displacement of said reflective surface out of saidplane substantially parallel with the z axis, at locations where m=0, 1,2, . . . k−1.
 12. A interferometric measurement system for determiningthe tilt of a reflective object mounted to a support, said systemcomprising: an interferometer system having first, second, third andfourth laser beam generators, each capable of generating a laser beam tomeasure a distance between said first, second, third or fourthgenerator, respectively, and a reflective surface mounted to a support;and a controller for receiving inputs from said interferometer systemand determining a tilt of said reflective surface with respect to a zaxis, wherein the support has a generally planar surface that isgenerally perpendicular to the z axis but which may tilt with respectthereto, and wherein the reflective surface is in a plane substantiallyparallel with the z axis and longitudinally extends substantiallyparallel to an axis normal to the z axis; and wherein said controllerdetermines a tilt of said reflective surface at a location ka along thelongitudinally extending direction of said reflective surface accordingto the following formula: Δ(ka)=Φ((k+1)a)−Φ(ka)  where: Δ(ka) is ameasure of a displacement of said reflective surface out of said planesubstantially parallel with the z axis, at location ka; Φ(ka) is ameasure of tilt of said reflective surface measured by said second andfourth laser beams; and Φ((k+1)a) is a measure of tilt of saidreflective surface measured by said first and third laser beams.
 13. Theinterferometric measurement system of claim 12, wherein: θ(x) is ameasure of tilt of the support; s(x) is a measure of displacement, outof the plane substantially parallel with the z axis, of the reflectivesurface when z=0; t(x) is a measure of displacement, out of the planesubstantially parallel with the z axis, of the reflective surface whenz=−b; δ(x) is a measure of displacement of the support along the axisnormal to the z axis; a measurement value for the second laser beam whenthe reflective surface is at a position y=ka is determined byL2(ka)=s(ka)+δ(ka)−(a/2)θ(ka); a measurement value for the fourth laserbeam when the reflective surface is at a position y=ka is determined byL4(ka)=t(ka)+δ(ka)+(a/2)θ(ka); a measurement value for the first laserbeam when the reflective surface is at a position y=ka is determined byL1(ka)=s(ka+a)+δ(ka)−(a/2)θ(ka); a measurement value for the third laserbeam when the reflective surface is at a position y=ka is determined byL3(ka)=t(ka+a)+δ(ka)+(a/2)θ(ka); Δ(ka)=J2(ka)−J1(ka)=Φ((k+1)a)−Φ(ka),where J1(ka)≡(L4(ka)−L2(ka))/b=(t(ka)−s(ka))/b+θ(ka)=Φ(ka)+θ(ka), andJ2(ka)≡(L3(ka)−L1(ka))/b=(t((k+1)a)−s((k+1)a))/b+θ(ka)=Φ((k+1)a)+θ(ka);the reflective surface can be moved to a position y=ka+a along the axisnormal to the z axis along which the reflective surface extendslongitudinally; a measurement value for the second laser beam when thereflective surface is at a position y=ka+a is determined byL2(ka+a)=s(ka+a)+δ(ka+a)−(a/2)θ(ka+a); a measurement value for thefourth laser beam when the reflective surface is at a position y=ka+a isdetermined by L4(ka+a)=t(ka+a)+δ(ka+a)+(a/2)θ(ka+a); a measurement valuefor the first laser beam when the reflective surface is at a positiony=ka+a is determined by L1(ka+a)=s(ka+a+a)+δ(ka+a)−(a/2)θ(ka+a); ameasurement value for the third laser beam when the reflective surfaceis at a position y=ka+a is determined byL3(ka+a)=t(ka+a+a)+δ(ka+a)+(a/2)θ(ka+a);Δ((k+1)a)=J2((k+1)a)−J1((k+1)a)=Φ((k+2)a)−Φ((k+1)a), whereJ1((k+1)a)≡(L4((k+1)a)−L2((k+1)a))/b=(t((k+1)a)−s((k+1)a))/b+θ((k+1)a)=Φ((k+1)a)+θ((k+1)a),andJ2((k+1)a)≡(L3((k+1)a)−L1((k+1)a))/b=(t((k+2)a)−s((k+2)a))/b+θ((k+1)a)=Φ((k+2)a)+θ((k+1)a);the reflective surface can be incrementally moved to additionalpositions in multiples of a along the axis normal to the z axis alongwhich the reflective surface extends longitudinally and additionalmeasurement values for the laser beams can be determined to arrive at aset of measurement values {Δ((k−1)a)=Φ(ka)−Φ((k−1)a);Δ((k−2)a)=Φ((k−1)a)−Φ((k−2)a); . . . ; Δ(0)=Φ(a)−Φ(0)}; and thecontroller determines a summation of the set of measurement values as${\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}} = {{\Phi ({ka})} - {\Phi (0)}}$

and a tilt of the reflective surface at a location ka along thelongitudinally extending direction of the reflective surface as${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta \quad {({ma}).}}}}$


14. The interferometric measurement system of claim 12, furthercomprising at least one motor operatively mounted to said support tomove said support and said reflective surface in directions along whichsaid reflective surface longitudinally extends, wherein said motorincrementally moves said reflective surface to measure displacement ofsaid reflective surface out of said plane substantially parallel withthe z axis at each incremental location.
 15. The interferometricmeasurement system of claim 14, wherein said tilt of said reflectivesurface, at a position ka is defined by:${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta \quad ({ma})}}}$

where: Φ(ka) is a measure of tilt of said reflective surface at positionka; Φ(0) is a measure of tilt of said reflective surface at an initialmeasurement location near one end of said reflective surface; and Δ(ma)is a measure of displacement of said reflective surface out of saidplane substantially parallel with the z axis, at locations where m=0, 1,2, . . . k−1; and a=the distance between any of the first and secondlaser beams, the third and fourth laser beams, the second and fourthlaser beams, and the first and third laser beams.
 16. Theinterferometric measurement system of claim 14, wherein said reflectivesurface comprises a first reflective surface; said interferometricmeasurement system further comprising: a second interferometer systemhaving fifth, sixth, seventh and eighth laser beam generators, eachcapable of generating a laser beam to measure a distance between saidfifth, sixth, seventh or eighth generator, respectively, and a secondreflective surface of a second reflective object mounted to the support,said reflective surface comprising a second reflective surface; whereinsaid controller receives inputs from said second interferometer systemand determines a tilt of the second reflective surface with respect tothe z axis, wherein the second reflective object is mounted to thesupport so that said second reflective surface is in a second planesubstantially parallel with the z axis and longitudinally extendssubstantially parallel to an axis normal to the z axis and normal to theaxis which said first reflective surface extends substantially parallelto; and wherein said controller determines a tilt of said secondreflective surface at a location ka along the longitudinally extendingdirection of said second reflective surface according to the followingformula: Δ(ka)=Φ((k+1)a)−Φ(ka)  where: Δ(ka) is a measure of adisplacement of said second reflective surface out of said second planesubstantially parallel with the z axis, at location ka; Φ(ka) is ameasure of tilt of said second reflective surface measured by said sixthand eighth laser beams; and Φ((k+1)a) is a measure of tilt of saidsecond reflective surface measured by said fifth and seventh laserbeams.
 17. The measurement system of claim 16, wherein: θ(y) is ameasure of tilt of the support; s(y) is a measure of displacement, outof the plane substantially parallel with the z axis, of the secondreflective surface when z=0; t(y) is a measure of displacement, out ofthe plane substantially parallel with the z axis, of the secondreflective surface when z=−b; δ(y) is a measure of displacement of thesupport along the axis normal to the z axis; a measurement value for thesixth laser beam when the second reflective surface is at a positionx=ka is determined by L6(ka)=s(ka)+δ(ka)−(a/2)θ(ka); a measurement valuefor the eighth laser beam when the second reflective surface is at aposition x=ka is determined by L8(ka)=t(ka)+δ(ka)+(a/2)θ(ka); ameasurement value for the fifth laser beam when the second reflectivesurface is at a position x=ka is determined byL5(ka)=s(ka+a)+δ(ka)−(a/2)θ(ka); a measurement value for the seventhlaser beam when the second reflective surface is at a position x=ka isdetermined by L7(ka)=t(ka+a)+δ(ka)+(a/2)θ(ka);Δ(ka)=J2(ka)−J1(ka)=Φ((k+1)a)−Φ(ka), whereJ1(ka)−(L8(ka)−L6(ka))/b=(t(ka)−s(ka))/b+θ(ka)=θ(ka)+θ(ka), andJ2(ka)=(L7(ka)−L5(ka))/b=(t((k+1)a)−s((k+1)a))/b+θ(ka)=Φ((k+1)a)+θ(ka);the second reflective surface can be moved to a position y=ka+a alongthe axis normal to the z axis along which the second reflective surfaceextends longitudinally; a measurement value for the sixth sensor whenthe second reflective surface is at a position x=ka+a is determined byL6(ka+a)=s(ka+a)+δ(ka+a)−(a/2)θ(ka+a); a measurement value for theeighth sensor when the second reflective surface is at a position x=ka+ais determined by L8(ka+a)=t(ka+a)+δ(ka+a)+(a/2)θ(ka+a); a measurementvalue for the fifth sensor when the second reflective surface is at aposition x=ka+a is determined byL5(ka+a)=s(ka+a+a)+δ(ka+a)−(a/2)θ(ka+a); a measurement value for theseventh sensor when the second reflective surface is at a positionx=ka+a is determined by L7(ka+a)=t(ka+a+a)+δ(ka+a)+(a/2)θ(ka+a);Δ((k+1)a)=J2((k+1)a)−J1((k+1)a)=Φ((k+2)a)−Φ((k+1)a), whereJ1((k+1)a)≡(L8((k+1)a)−L6((k+1)a))/b=(t((k+1)a)−s((k+1)a))/b+θ((k+1)a)=Φ((k+1)a)+θ((k+1)a),andJ2((k+1)a)=(L7((k+1)a)−L5((k+1)a))/b=(t((k+2)a)−s((k+2)a))/b+θ((k+1)a)=Φ((k+2)a)+θ((k+1)a);the second reflective surface can be incrementally moved to additionalpositions in multiples of a along the axis normal to the z axis alongwhich the second reflective surface extends longitudinally andadditional measurement values for the sensors can be determined toarrive at a set of measurement values {Δ((k−1)a)=Φ(ka)−Φ((k−1)a);Δ((k−2)a)=Φ((k−1)a)−Φ((k−2)a); . . . ; Δ(0)=Φ(a)−Φ(0)}; and thecontroller determines a summation of the set of measurement values as${\sum\limits_{m = 0}^{k - 1}{\Delta \quad ({ma})}} = {{\Phi ({ka})} - {\Phi (0)}}$

and a tilt of the second reflective surface at a location ka along thelongitudinally extending direction of the second reflective surface as${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta \quad {({ma}).}}}}$


18. The interferometric measurement system of claim 16, furthercomprising at least one second motor operatively mounted to said supportto move said support and said second reflective surface in directionsalong which said second reflective surface longitudinally extends,wherein said at least one second motor incrementally moves said secondreflective surface to measure displacement of said second reflectivesurface out of said second plane substantially parallel with the z axisat each incremental location.
 19. The interferometric measurement systemof claim 18, wherein said tilt of said second reflective surface, at aposition ka is defined by:${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta \quad ({ma})}}}$

where: Φ(ka) is a measure of tilt of said second reflective surface atposition ka; Φ(0) is a measure of tilt of said second reflective surfaceat an initial measurement location near one end of said secondreflective surface; and Δ(ma) is a measure of displacement of saidsecond reflective surface out of said second plane substantiallyparallel with the z axis, at locations where m=0, 1, 2, . . . k−1; anda=the distance between any of the fifth and sixth laser beams, theseventh and eighth laser beams.
 20. A method of measuring the tilt of asubstantially planar surface with respect to a vertical axis,comprising: providing a measurement system having the capability ofmeasuring distances between first, second, third and fourth adjacentlocations on the substantially planar surface and respective first,second, third and fourth adjacent locations on the measurement system,where the distances measured are along imaginary lines substantiallyperpendicular to the substantially planar surface; positioning thesubstantially planar surfaces such that the measurement system is nearan end of the substantially planar surface; measuring distances betweenthe pairs of first, second, third and fourth locations; subtracting thedistance between the second locations from the distance between thefourth locations and dividing the difference by a distance between thesecond and fourth locations on the substantially planar surface to givea term J1; subtracting the distance between the first locations from thedistance between the third locations and dividing the difference by thedistance between the first and third locations on the substantiallyplanar surface to give a term J2; and determining a tilt of thesubstantially planar surface at the location of the substantially planarsurface according to the following formula:Δ(ka)=J2(ka)−J1(ka)=Φ((k+1)a)−Φ(ka)  where: Δ(ka) is a measure of adisplacement out of the substantially planar surface; Φ(ka) is a measureof tilt of the substantially planar surface with respect to the verticalaxis measured between the second and fourth locations; Φ((k+1)a) is ameasure of tilt of the substantially planar surface with respect to thevertical axis measured between the first and third locations; and a is adistance between the first and second locations on the substantiallyplanar surface.
 21. The method of claim 20, further comprising:incrementally moving the substantially planar surface in a directionparallel to an axis normal to the vertical axis and away from the end ofthe surface by the distance a; measuring distances between the first,second, third and fourth locations on the measurement system and therespective four new locations on the substantially planar surface;subtracting the distance between the second locations from the distancebetween the fourth locations and dividing the difference by a distancebetween the second and fourth locations on the substantially planarsurface to give a term J1; subtracting the distance between the firstlocations from the distance between the third locations and dividing thedifference by the distance between the first and third locations on thesubstantially planar surface to give a term J2; and determining a tiltof the substantially planar surface at the new location incrementallyremoved from a previously measured location according to the followingformula: Δ(ka+a)=J 2(ka+a)−J 1(ka+a)=Φ((k+2)a)−Φ((k+1)a)  where: Δ(ka+a)is a measure of a displacement out of the substantially planar surface;Δ(k+1)a) is a measure of tilt of the substantially planar surface withrespect to the vertical axis measured between the second and fourthlocations on the measurement system and the new locations on thesubstantially planar surface; and Φ((k+2)a) is a measure of tilt of thesubstantially planar surface with respect to the vertical axis measuredbetween the first and third locations on the measurement system and thenew locations on the substantially planar surface.
 22. The method ofclaim 21, further comprising incrementally repeating the method of claim20 until an opposite end of the substantially planar surface is reachedand no further incremental measurements can be taken, or until apredetermined length of the substantially planar surface has beenmeasured.
 23. The method of claim 22, further comprising determining atilt of the substantially planar surface with respect to the verticalaxis for any predetermined position ka according to the followingformula:${\Phi ({ka})} = {{\Phi (0)} + {\sum\limits_{m = 0}^{k - 1}{\Delta ({ma})}}}$

where: Φ(ka) is a measure of tilt of the substantially planar surfacewith respect to the vertical axis at position ka; Φ(0) is a measure oftilt of the substantially planar surface near the end of thesubstantially planar surface where an initial measurement was taken inclaim 19; and Δ(ma) is a measure of displacement out of saidsubstantially planar surface, at locations where m=0, 1, 2, . . . k−1.24. A measurement system for determining the tilt of a reflective objectmounted to a support, the system comprising: first, second, third andfourth sensors, each capable of generating data indicative of a distancebetween the first, second, third or fourth sensor, respectively, and areflective surface of the reflective object; and a controller forreceiving inputs from the first, second, third and fourth sensors anddetermining a tilt of the reflective surface with respect to a z axis; wherein: the support has a generally planar surface that is generallyperpendicular to the z axis but which may tilt with respect thereto, thereflective object is mounted to the support so that the reflectivesurface is in a plane substantially parallel with the z axis andlongitudinally extends substantially parallel to an axis normal to the zaxis; the first and second sensors are aligned substantially parallel tothe axis normal to the z axis along which the reflective surface extendslongitudinally and are separated by a distance a; the third and fourthsensors are aligned substantially parallel to the axis normal to the zaxis along which the reflective surface extends longitudinally and areseparated by the distance a; the first and third sensors are alignedsubstantially parallel to the z axis and are separated by a distance b;the second and fourth sensors are aligned substantially parallel to thez axis and are separated by the distance b; and the controllerdetermines a tilt of the reflective surface along the longitudinallyextending direction of the reflective surface.
 25. A method of measuringthe tilt of a substantially planar surface with respect to a verticalaxis, comprising: providing a measurement system having the capabilityof measuring distances between first, second, third and fourth adjacentlocations on the substantially planar surface and respective first,second, third and fourth adjacent locations on the measurement system,where the distances measured are along imaginary lines substantiallyperpendicular to the substantially planar surface; positioning thesubstantially planar surfaces such that the measurement system is nearan end of the substantially planar surface; measuring distances betweenthe pairs of first, second, third and fourth locations; subtracting thedistance between the second locations from the distance between thefourth locations and dividing the difference by a distance between thesecond and fourth locations on the substantially planar surface to givea term J1; subtracting the distance between the first locations from thedistance between the third locations and dividing the difference by thedistance between the first and third locations on the substantiallyplanar surface to give a term J2; and determining a tilt of thesubstantially planar surface at the location of the substantially planarsurface.